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A123922
Number of 2143-avoiding Dumont paths of the 2nd kind of length 2n.
0
1, 1, 2, 6, 21, 84, 360, 1650, 7865, 39039, 198744, 1039584, 5534928, 30046752, 165257136, 922280634, 5199131025, 29644168125, 170375955750, 988180543350, 5768664340725, 33927954699600, 200617471267200, 1193673954039840
OFFSET
0,3
LINKS
A. Burstein, S. Elizalde and T. Mansour, Restricted Dumont Permutations, Dyck Paths and Noncrossing Partitions, arXiv:math/0610234 [math.CO], 2006.
FORMULA
a(n) = A047749(n)*A047749(n+1).
Conjecture: 16*n*(n+2)*(n+1)^2*a(n) -108*n*(n+1)*(2*n-1)*a(n-1) -9*(3*n-5)*(3*n-1)*(3*n-4)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Jan 25 2013
EXAMPLE
For n=2, there are 3 Dumont permutations of the 2nd kind of length 2n=4, namely {2143,3142,4132}.
Avoiding 2143, the cardinality of this set is reduced to a(2)=2.
MATHEMATICA
b[n_] := If[EvenQ[n], Binomial[3n/2, n/2]/(n+1), Binomial[(3n-1)/2, (n+1)/2 ]/n];
a[n_] := b[n] b[n+1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jul 27 2018 *)
PROG
(PARI) A047749(n)={ my(m=floor(n/2)); if(n % 2, binomial(3*m+1, m+1)/(2*m+1), binomial(3*m, m)/(2*m+1)); }
a(n)={ A047749(n)*A047749(n+1); }
CROSSREFS
Sequence in context: A150223 A150224 A150225 * A350798 A326276 A099947
KEYWORD
easy,nonn
AUTHOR
R. J. Mathar, Nov 20 2006
STATUS
approved